1. Solving Linear Systems (in three variables)
Section 1.4 beginning on page 30

2. A Linear Equation in Three Variables

3. What Does This Look Like?

4. Solving Algebraically

5. Examples Solve each system:
(−1,5,2)
𝑁𝑜 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛
Infinitely Many Solutions
(𝑥,𝑥+3,0)

6. Real Life Problem
Example 4: An amphitheater charges $75 for each seat in Section A, $55 for each seat in Section B, and $30 for each lawn seat. There are three times as many seats in Section B as in Section A. The revenue from selling all 23,000 seats is $870,000. How many seats are in each section of the amphitheater?
Declare your variables: X = Number of seats in Section A
Y = Number of seats in Section B
Z = Number of seats on the Lawn
Write three equations using the information provided:
One equation can represent the relationship between the number of seats in section b and section a
One equation can represent the total number of seats
One equation can represent the revenue from selling out
𝑦=3𝑥 𝑥+𝑦+𝑧=23,000 75𝑥+55𝑦+30𝑧=870,000
There are 1500 seats in Section A, 4500 seats in Section B, and lawn seats.

7. Monitoring Progress Solve the system. 𝑥−2𝑦+𝑧=−11 3𝑥+2𝑦−𝑧=7 −𝑥+2𝑦+4𝑧=−9
2) 𝑥+𝑦−𝑧=−1 4𝑥+4𝑦−4𝑧=−2 3𝑥+2𝑦+𝑧=0
3) 𝑥+𝑦+𝑧=8 𝑥−𝑦+𝑧=8 2𝑥+𝑦+2𝑧=16
(−1,3,−4)
𝑁𝑜 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛
(8−𝑧,0,𝑧)